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Is there a musical theory of harmony or a system of chord-understanding that does not rely on the 12-tone temperament?

For example, a fretless instrument has (theoretically) an infinity of tones between one note and the next. Thus, the concept of note becomes blurry and the point of interest becomes the interval (or frequency-distance) between tones.

Therefore, I am curious if there is a (more-or-less magical) conjoined theory that works on a continuous (or non-quantized temperament).

If not, is there any research or have there been past attempts at a system ?

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    Look up Microtonal music. The subject is too vast for a real answer. I offer the music of Harry Partch as one of many many varied and vastly different examples along this subject. This includes some music that is ethnomusicological as well as new made up systems and theories of music. – amalgamate Mar 25 '16 at 14:15
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It depends on how you define the 12-tone octave.

Up until at least the 19th century, several theories did not use equal temperament to conceptualize musical space. Imagine that you ascend from C by perfect fifth; if you do this twelve times in equal temperament, you will end on a C that is perfectly in tune with the original starting C. If, however, you do this in just intonation, your final C will be considerably higher than your original C. In this sense, these theories did in fact have "an infinity of tones between one note and the next." As a good starter on this, maybe check out the Tonnetze by Oettingen and Riemann.

More recently, you can look at analyses and theories of microtonal and spectral music.

But in both cases, these are based off of the 12-tone octave, just with gradations between the pitches.

In terms of a "more-or-less magical" conjoined theory...I'm not sure I can help there.

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theory that works on a continuous (or non-quantized octave).

In musique concrète, the theoretical framework is audio waveforms, not octaves or tunings or scales. You’re not limited to a system of writing down musical notes so that the next person can play them on a standard instrument with a standard octave and standard tuning, because you write down musique concrète with an audio recorder. You just write down the audio itself.

a fretless instrument has (theoretically) an infinity of tones between one note and the next. Thus, the concept of note becomes blurry

That is a musique concrète way of looking at a fretless instrument: based on the infinity of sounds you can make with it, not the particular set of standard notes or tunings you can use it to play from a score. You could also rattle the bass, strike the body of the bass, interfere electromagnetically with the pickups, use any number of effects processors. Anything that makes a sound that you can record.

Musique concrète — Wikipedia

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    Musique concrète is using everyday sounds to compose music, you wouldn't use recordings of instruments as that would defeat the purpose of it. Also it's a composition technique not a theoretical system. – Dom♦ Mar 25 '16 at 11:18
  • @Dom I got the impression that any sound is valid in a concrète piece, including vocal or instrumental sounds (e.g. the piano and singing in Cinq études de bruits by Pierre Schaeffer). I thought the key point of concrète is an emphasis on the music being defined by actual ('concrete') recorded sound, rather than the tradition of instrumentalists rendering a (definitive) score. – topo Reinstate Monica Mar 25 '16 at 12:02
  • @topomorto you wouldn't just throw samples of a song together and call it musique concrète. The recording aspect of it is why it is named that way, but the point was to use the recordings so you could have access to sounds you wouldn't have any other way espically from a performance stand point. There may be typical instruments in some musique concrète pecies, but that is not the focus of it. – Dom♦ Mar 25 '16 at 12:14
  • @Dom true that instruments aren't a focus of music concrete - I was just saying that it's going a bit far to say that you wouldn't use recordings of instruments. – topo Reinstate Monica Mar 25 '16 at 13:15
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The octave can be split into more intervals than 12. There is 19 equal temperament, and other temperaments based on octave division into 31, 41 or 53 equal intervals.

Some 30 years ago I had to work on a mathematical problem where it was proved that some divisions were "better" than others. Better in the sense that they better approximated simple fractions.

The ear likes chords composed of notes with frequencies in simple (fractional) ratios. Hence the traditional division into 12 equal parts comes from the fact that 2^(7/12) is a good approximation of 3/2 (perfect fifth), 2^(4/12) an approximation of 5/4 (major third), and 2^(3/12) an approximation of 6/5 (minor third).

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  • As you say. But of course it's a tradeoff between how good the intervals you need (fifths, fourths, thirds) sound and how many divisions you need. Twelve tone equal temperament has great fifths but only mediocre thirds; 53 TET has great fifths and thirds but a lot of notes. – Scott Wallace May 10 '16 at 14:41

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